Here,

2 cos\(\theta\) = 1

or, cos\(\theta\) = \(\frac 12\)

or, cos\(\theta\) = cos 60° , cos (360° - 60°)

∴ \(\theta\) = 60° , 300° _Ans

Here,

3 tan\(\theta\) = \(\sqrt 3\)

or, tan\(\theta\) = \(\frac {\sqrt 3}3\)

or, tan\(\theta\) = \(\frac {\sqrt 3}{\sqrt 3 × \sqrt 3}\)

or, tan\(\theta\) = \(\frac 1{\sqrt 3}\)

or, tan\(\theta\) = tan 30° , tan (180° + 30°)

∴ \(\theta\) = 30° , 210° _Ans

Here,

\(\sqrt 2\) sec\(\theta\) + 2 = 0

or, \(\sqrt 2\) sec\(\theta\) = - 2

or. sec\(\theta\) = \(\frac {-2}{\sqrt 2}\)

or, sec\(\theta\) = -\(\frac {\sqrt 2 × \sqrt 2}{\sqrt 2}\)

or, sec\(\theta\) = -\(\sqrt 2\)

or, sec \(\theta\) = sec (180° - 45°), sec (180v + 45°)

∴ \(\theta\) = 135° , 225° _Ans

Here,

tanx - sinx = 0

or, \(\frac {sinx}{cosx}\) - sinx = 0

or, \(\frac {sinx - sinx cosx}{cosx}\) = 0

or, sinx (1 - cosx) = 0

Either,

sinx = 0

sinx = sin 0°

∴ x = 0°

Or,

1 - cosx = 0

or, cosx= 1

or, cosx = cos180°

∴ x = 180°

∴ x = 0° , 180° _Ans

Here,

2 sin\(\theta\) + 1 = 0

or, 2 sin\(\theta\) = - 1

or, sin\(\theta\) = -\(\frac 12\)

or, sin\(\theta\) = sin (180° + 30°), sin (360° - 30°)

∴ \(\theta\) = 210°, 330° _Ans

Here,

cosx + secx = \(\frac 52\)

or, cosx + \(\frac 1{cosx}\) = \(\frac 52\)

or, \(\frac {cos^2x + 1}{cosx}\) = \(\frac 52\)

or, 2 cos²x + 2 = 5 cosx

or, 2 cos²x - 5 cosx+ 2 = 0

or, 2 cos²x - 4 cosx - cosx + 2 = 0

or, 2 cosx (cosx - 2) - 1 (cosx - 2) = 0

or, (cosx - 2) (2 cosx - 1) = 0

Either,

cosx - 2 = 0

or, cosx = 2 (Impossible)

Or,

2 cosx - 1 = 0

or, 2 cosx = 1

or, cosx = \(\frac 12\)

or, cosx = cos 60°

∴ x = 60° _Ans

Here,

4 sin\(\alpha\) = 3 cosec\(\alpha\)

or, 4 sin\(\alpha\) = 3\(\frac 1{sin\alpha}\)

or, 4 sin²\(\alpha\) = 3

or, sin²\(\alpha\) = \(\frac 34\)

∴ sin\(\alpha\) =± \(\frac {\sqrt 3}2\)

Taking positive,

sin\(\alpha\) = \(\frac {\sqrt 3}2\)

or, sin\(\alpha\) = sin 60° , sin (180° - 60°)

∴ \(\alpha\) = 60° , 120°

Taking negative,

sin\(\alpha\) = -\(\frac {\sqrt 3}2\)

or, sin\(\alpha\) = sin (180° + 60°), sin (360° - 60°)

∴ \(\alpha\) = 240° , 300°

∴ \(\alpha\) = 60°, 120°, 240°, 300° _Ans

Here,

cos²\(\theta\) - 1 = sin\(\theta\)

or, 1 - sin²\(\theta\) - 1 - sin\(\theta\) = 0

or, - sin²\(\theta\) - sin\(\theta\) = 0

or, - sin\(\theta\) (sin\(\theta\) - 1) = 0

Either,

sin\(\theta\) = 0

or, sin\(\theta\) = sin 0°, sin 180°

Or,

sin\(\theta\) - 1 = 0

or, sin\(\theta\) = 1

or, sin\(\theta\) = sin 90°

∴ \(\theta\) = 0° , 90° , 180° _Ans

Here,

sin\(\theta\) - cos\(\theta\) = 0

or, sin\(\theta\) = cos\(\theta\)

or, \(\frac {sin\theta}{cos\theta}\) = 1

or, tan\(\theta\) = tan 45°

∴ \(\theta\) = 45° _Ans

Here,

cot²x + cosec²x = 3

or, cosec²x - 1 + cosec²x = 3

or, 2 cosec²x = 3 + 1

or, cosec²x = \(\frac 42\)

or, cosec²x = 2

or, cosec x =± \(\sqrt 2\)

For positive sign,

cosec x = cosec 45° , cosec (360° - 45°)

For negative sign,

cosec x = cosec (180° - 45°) , cosec (180° + 45°)

∴ x = 45°, 135°, 225° and 315° _Ans

Here,

cos 2x = sin x

or, cos 2x = cos (90° - x)

or, 2x = 90° - x

or, 3x = 90°

or, x = \(\frac {90°}3\)

∴ x = 30° _Ans

Here,

sin\(\theta\) + cos\(\theta\) = \(\sqrt 2\)

Squaring on both sides:

(sin\(\theta\) + cos\(\theta\))²= (\(\sqrt 2\))²

or, sin²\(\theta\) + cos²\(\theta\) + 2 sin\(\theta\) cos\(\theta\) = 2

or, 1 + sin 2\(\theta\) = 2

or, sin 2\(\theta\) = 2 - 1

or, sin 2\(\theta\) = 1

or, sin 2\(\theta\) = sin 90°

or, 2\(\theta\) = 90°

or, \(\theta\) = \(\frac {90°}2\)

∴ \(\theta\) = 45° _Ans

Here,

2 cos²\(\theta\) = - \(\sqrt 3\) cos\(\theta\)

or, 2 cos²\(\theta\) + \(\sqrt 3\) cos\(\theta\) = 0

or, cos\(\theta\) (2 cos\(\theta\) + \(\sqrt 3\)) = 0

Either,

cos\(\theta\) = 0

or, cos\(\theta\) = cos 90°

Or,

2 cos\(\theta\) + \(\sqrt 3\) = 0

or, 2 cos\(\theta\) = - \(\sqrt 3\)

or, cos\(\theta\) = - \(\frac {\sqrt 3}2\)

or, cos\(\theta\) = cos (180° - 30°)

∴ \(\theta\) = 90° and 150° _Ans

Here,

cos²\(\frac {\theta}2\) - cos\(\frac {\theta}2\) + \(\frac 14\) = 0

or, (cos\(\frac {\theta}2\))² - 2 . cos\(\frac {\theta}2\) . (\(\frac 12\))² = 0

or, (cos\(\frac {\theta}2\) - \(\frac 12\))² = 0

or, cos\(\frac {\theta}2\) - \(\frac 12\) = 0

or, cos\(\frac {\theta}2\) = \(\frac 12\)

or, cos\(\frac {\theta}2\) = cos 60°

or, \(\frac {\theta}2\) = 60°

or, \(\theta\) = 2× 60°

∴ \(\theta\) = 120° _Ans

Here,

2 cos²\(\theta\) = 3 sin\(\theta\) = 0

or, 2 (1 - sin²\(\theta\)) - 3 sin\(\theta\) = 0

or, 2 - 2 sin²\(\theta\) - 3 sin\(\theta\) = 0

or, - 2 sin²\(\theta\) - 3 sin\(\theta\) + 2 = 0

or, - (2 sin²\(\theta\) + 3 sin\(\theta\) - 2) = 0

or, 2 sin²\(\theta\) + 4 sin\(\theta\) - sin\(\theta\) - 2 = 0

or, 2 sin\(\theta\) (sin\(\theta\) + 2) - 1 (sin\(\theta\) + 2) = 0

or, (sin\(\theta\) + 2) (2 sin\(\theta\) - 1) = 0

Either,

sin\(\theta\) + 2 = 0

or, sin\(\theta\) = - 2 (Impossible)

Or,

2 sin\(\theta\) - 1 = 0

or, 2 sin\(\theta\) = 1

or, sin\(\theta\) = \(\frac 12\)

or, sin\(\theta\) = sin 30° , sin (180° - 30°)

∴ \(\theta\) = 30°, 150° _Ans

Here,

cos²\(\theta\) - sin\(\theta\) = \(\frac 14\)

or, 4 (1 - sin²\(\theta\) - sin\(\theta\)) = 1

or, 4 - 4 sin²\(\theta\) - 4 sin\(\theta\) - 1 = 0

or, - (4 sin²\(\theta\) + 4 sin\(\theta\) - 3) = 0

or, 4 sin²\(\theta\) + 6 sin\(\theta\) - 2 sin\(\theta\) - 3 = 0

or, 2 sin\(\theta\) (2 sin\(\theta\) + 3) - 1 (2 sin\(\theta\) + 3) = 0

or, (2 sin\(\theta\) + 3) (2 sin\(\theta\) - 1) = 0

Either,

2 sin\(\theta\) + 3 = 0

or, 2 sin\(\theta\) = -3

or, sin\(\theta\) = -\(\frac 32\)

sin\(\theta\) is not less than -1. So, it is impossible.

Or,

2 sin\(\theta\) - 1 = 0

or, 2 sin\(\theta\) = 1

or, sin\(\theta\) = \(\frac 12\)

or, sin\(\theta\) = sin 30°, sin (180° - 30°)

∴ \(\theta\) = 30° and 150° _Ans

Here,

sec\(\theta\) . tan\(\theta\) = \(\sqrt 2\)

or, \(\frac 1{cos\theta}\) . \(\frac {sin\theta}{cos\theta}\) = \(\sqrt 2\)

or, \(\frac {sin\theta}{cos^2\theta}\) = \(\sqrt 2\)

or, \(\frac {sin\theta}{1 - sin^2\theta}\) = \(\sqrt 2\)

or, sin\(\theta\) = \(\sqrt 2\) - \(\sqrt 2\) sin²\(\theta\)

or, \(\sqrt 2\) sin²\(\theta\) + sin\(\theta\) - \(\sqrt 2\) = 0

or, \(\sqrt 2\) sin²\(\theta\) + 2 sin\(\theta\) - sin\(\theta\) - \(\sqrt 2\) = 0

or, \(\sqrt 2\) sin\(\theta\) (sin\(\theta\) + \(\sqrt 2\)) - 1 (sin\(\theta\) + \(\sqrt 2\)) = 0

or, (sin\(\theta\) + \(\sqrt 2\)) (\(\sqrt 2\) sin\(\theta\) - 1) = 0

Either,

sin\(\theta\) + \(\sqrt 2\) = 0

or, sin\(\theta\) = -\(\sqrt 2\) (Impossible)

Or,

\(\sqrt 2\) sin\(\theta\) - 1 = 0

or, \(\sqrt 2\) sin\(\theta\) = 1

or, sin\(\theta\) = \(\frac 1{\sqrt 2}\)

or, sin\(\theta\) = sin 45° , sin (180° - 45°)

∴ \(\theta\) = 45° , 135° _Ans

Here,

1 + cos\(\theta\) = 2 sin²\(\theta\)

or, 2 sin²\(\theta\) - cos\(\theta) - 1 = 0

or, 2 (1 - cos²\(\theta\)) - cos\(\theta\) - 1 = 0

or, 2 - 2 cos²\(\theta\) - cos\(\theta\) - 1 = 0

or, - 2 cos²\(\theta\) - cos\(\theta\) + 1 = 0

or, - (2 cos²\(\theta\) + cos\(\theta\) - 1) = 0

or, 2 cos²\(\theta\) + 2 cos\(\theta\) - cos\(\theta\) - 1 = 0

or, 2 cos\(\theta\) (cos\(\theta\) + 1) - 1 (cos\(\theta\) + 1) = 0

or, (cos\(\theta\) + 1) (2 cos\(\theta\) - 1) = 0

Either,

cos\(\theta\) + 1 = 0

cos\(\theta\) = - 1

cos\(\theta\) = cos 180°

Or,

2 cos\(\theta\) - 1 = 0

2 cos\(\theta\) = 1

cos\(\theta\) = \(\frac 12\)

cos\(\theta\) = cos 60°

∴ \(\theta\) = 60° and 180° _Ans

Here,

cos 2\(\theta\) = sin\(\theta\)

or, 1 - 2 sin²\(\theta\) = sin\(\theta\)

or, 2 sin²\(\theta\) + sin\(\theta\) - 1 = 0

or, 2 sin²\(\theta\) + 2 sin\(\theta\) - sin\(\theta\) - 1 = 0

or, 2 sin\(\theta\) (sin\(\theta\) + 1) - 1 (sin\(\theta\) + 1) = 0

or, (sin\(\theta\) + 1) (2 sin\(\theta\) - 1) = 0

Either,

sin\(\theta\) + 1 = 0

sin\(\theta\) = - 1

sin\(\theta\) = sin 270°

Or,

2 sin\(\theta\) - 1 = 0

2 sin\(\theta\) = 1

sin\(\theta\) = \(\frac 12\)

sin\(\theta\) = sin 30°, sin (180° - 30°)

∴ \(\theta\) = 30°, 150°, 270° _Ans

Here,

2 cos²\(\theta\) + sin\(\theta\) = 2

or, 2 (1 - sin²\(\theta\)) + sin\(\theta\) - 2 = 0

or, 2 - 2 sin²\(\theta\) + sin\(\theta\) - 2 = 0

or, - 2 sin²\(\theta\) + sin\(\theta\) = 0

or, - sin\(\theta\) (2 sin\(\theta\) - 1) = 0

or, sin\(\theta\) (2 sin\(\theta\) - 1) = 0

Either,

- sin\(\theta\) = 0

sin\(\theta\) = sin 0°, sin 180°

Or,

2 sin\(\theta\) - 1 = 0

2 sin\(\theta\) = 1

sin\(\theta\) = \(\frac 12\)

sin\(\theta\) = sin 30°, sin (180° - 30°)

∴ \(\theta\) = 0°, 30°, 150° and 180° _Ans

Here,

tan²A - (1 + \(\sqrt 3\)) tanA + \(\sqrt 3\) = 0

or, tan²A - tanA - \(\sqrt 3\) tanA + \(\sqrt 3\) = 0

or, tanA (tanA - 1) - \(\sqrt 3\) (tanA - 1) = 0

or, (tanA - 1) (tanA - \(\sqrt 3\)) = 0

Either,

tanA - 1 = 0

tanA = 1

tanA = tan 45° , tan (180° + 45°)

Or,

tanA - \(\sqrt 3\) = 0

tanA = \(\sqrt 3\)

tanA = tan 60°, tan (180° + 60°)

∴ A = 45°, 60°, 225°, 240° _Ans

Here,

cos²x = 3 sin²x + 4 cosx

or, cos²x - 3 sin²x - 4 cosx = 0

or, cos²x - 3(1 - cos²x) - 4 cosx = 0

or, cos²x - 3 + 3 cos²x - 4 cosx = 0

or, 4 cos²x - 4 cosx- 3 = 0

or, 4 cos²x - 6 cosx + 2 cosx - 3 = 0

or, 2 cosx(2 cosx - 3) + 1 (2 cosx - 3) = 0

or, (2 cos x- 3) (2 cos x + 1) = 0

Either,

2 cos x - 3 = 0

2 cos x = 3

cos x = \(\frac 32\) (Impossible)

Or,

2 cos x+ 1 = 0

2 cos x = - 1

cos x= -\(\frac 12\)

cos x = cos (180° - 60°), cos (180° + 60°)

∴ x = 120° , 240° _Ans

Here,

\(\sqrt 3\) cotA = \(\frac {\sqrt 3}{sinA}\) - 1

or, \(\sqrt 3\)\(\frac {cosA}{sinA}\) = \(\frac {\sqrt 3- sinA}{sinA}\)

or, \(\sqrt 3\)cosA sinA = \(\sqrt 3\)sinA - sin²A

or, sin²A + \(\sqrt 3\)cosA sinA - \(\sqrt 3\)sinA = 0

or, sinA(sinA + \(\sqrt 3\)cosA - \(\sqrt 3\)) = 0

Either,

sinA = 0

sinA = sin 0° , sin 180°, sin 360°

A = 0°, 180°, 360° (These value does not satisfies in the given equation so we eliminate these values.)

or, sinA+ \(\sqrt 3\)cosA - \(\sqrt 3\) = 0

or, sinA + \(\sqrt 3\)cosA = \(\sqrt 3\)

or, \(\frac {sinA}{\sqrt {1^2 + \sqrt {(3)^2}}}\) + \(\frac {\sqrt 3 cosA}{\sqrt {1^2 + \sqrt {(3)^2}}}\) = \(\frac {\sqrt 3}{\sqrt {1^2 + \sqrt {(3)^2}}}\)

or, \(\frac {sinA}{\sqrt {1 + 3}}\) + \(\frac {\sqrt 3 cosA}{\sqrt {1 + 3}}\) = \(\frac {\sqrt 3}{\sqrt {1 + 3}}\)

or, \(\frac 12\)sinA + \(\frac {\sqrt 3}2\)cosA = \(\frac {\sqrt 3}2\)

or, cosA cos30° + sinA sin 30° = \(\frac {\sqrt 3}2\)

or, cos(A - 30°) = cos 30° , cos(360° - 30°)

or, cos (A - 30°) = cos 30°....................(i)

or, cos (A - 30°) = cos 330°..................(ii)

From (i)

A - 30° = 30°

A = 30° + 30°

A = 60°

From (ii)

A - 30° = 330°

A = 330° + 30°

A = 360°

Since, A = 360° does not satisfy in the given equation.

∴ A = 60° _Ans

Here,

sin 3x + sin x = 2 sin x

or, 2 sin x\(\frac {3x + x}2\) cos\(\frac {3x - x}2\) = 2 sinx

or, 2 sin2x cosx = 2 sinx

or, 2× 2 sinx cosxcosx - 2 sinx = 0

or, 2 sinx(2 cos²x - 1) = 0

Either,

2 sinx = 0

sinx = 0

sinx = sin 0° , sin 180°

Or,

2 cos²x - 1 = 0

2 cos²x = 1

cos²x = \(\frac 12\)

cosx=± \(\frac 1{\sqrt 2}\)

cosx = cos 45° , cos (180° - 45°)

∴ x = 0° , 45° , 135° , 180° _Ans

Here,

\(\frac {\sqrt 3}{sin 2\theta}\) + \(\frac 1{cos 2\theta}\) = 4

or, \(\frac {\sqrt 3 cos 2\theta + sin 2\theta}{sin 2\theta cos 2\theta}\) = 4

or, \(\frac {\frac {\sqrt 3}2 cos 2\theta + \frac 12 sin 2\theta}{2 sin 2\theta cos 2\theta}\) = \(\frac 44\)

or, \(\frac {sin 60° cos 2\theta + cos 60° sin 2\theta}{sin 4\theta}\) = 1

or, sin (60° + 2\(\theta\)) = sin 4\(\theta\)

or, 60° + 2\(\theta\) = 4\(\theta\)

or, 4\(\theta\) - 2\(\theta\) = 60°

or, 2\(\theta\) = 60°

or, \(\theta\) = \(\frac {60°}2\)

∴ \(\theta\) = 30° _Ans

Here,

sin 2\(\theta\) + sin 4\(\theta\) = cos\(\theta\) + cos 3\(\theta\)

or, 2 sin\(\frac {2\theta + 4\theta}2\) cos\(\frac {2\theta - 4\theta}2\) = 2 cos\(\frac {\theta + 3\theta}2\) cos\(\frac {\theta - 3\theta}2\)

or, 2 sin 3\(\theta\) . cos\(\theta\) = 2 cos 2\(\theta\) . cos\(\theta\)

or, 2 sin 3\(\theta\) . cos\(\theta\) - 2 cos 2\(\theta\) . cos\(\theta\) = 0

or, 2 cos\(\theta\) (sin 3\(\theta\) - cos 2\(\theta\)) = 0

Either,

2 cos\(\theta\) = 0

cos\(\theta\) = 0

cos\(\theta\) = cos 90°

Or,

sin 3\(\theta\) - cos 2\(\theta\) = 0

sin 3\(\theta\) = cos\(\theta\)

sin 3\(\theta\) = sin(90° - 2\theta)

3\(\theta\) = 90° - 2\(\theta\)

3\(\theta\) + 2\(\theta\) = 90°

5\(\theta\) = 90°

\(\theta\) = \(\frac {90°}5\)

\(\theta\) = 18°

∴ \(\theta\) = 18° , 90° _Ans

Here,

tan\(\theta\) + tan 2\(\theta\) + \(\sqrt 3\) tan\(\theta\) tan 2\(\theta\) = \(\sqrt 3\)

or, tan\(\theta\) + tan 2\(\theta\) = \(\sqrt 3\) -\(\sqrt 3\) tan\(\theta\) tan 2\(\theta\)

or, tan\(\theta\) + tan 2\(\theta\) = \(\sqrt 3\) (1 - tan\(\theta\) tan 2\(\theta\))

or, \(\frac {tan\theta + tan 2\theta}{1 - tan\theta tan 2\theta}\) = \(\sqrt 3\)

or, tan (\(\theta\) + 2\(\theta\)) = \(\sqrt 3\)

or, tan 3\(\theta\) = \(\sqrt 3\)

or, tan 3\(\theta\) = tan 60°, tan (180° + 60°)

Either,

3\(\theta\) = 60°

\(\theta\) = \(\frac {60°}3\)

\(\theta\) = 20°

Or,

3\(\theta\) = 240°

\(\theta\) = \(\frac {240°}3\)

\(\theta\) = 80°

∴ \(\theta\) = 20°, 80° _Ans

Here,

cot x + cot y = 2

or, \(\frac {cos x}{sin x}\) + \(\frac {cos y}{sin y}\) = 2

or, \(\frac {cos x sin y + cos y sin x}{sin x sin y}\) = 2

or, sin x cos y + cos x sin y = 2 sin x cos y

or, sin (x + y) = \(\frac {\sqrt 3}2\) [\(\because\) 2 sin x cos y = \(\frac {\sqrt 3}2\)]

or, sin (x + y) = sin 60°, sin (180° - 60°)

∴ x + y = 60°, 120° _Ans

Here,

\(\sqrt 2\) sin\9\theta\) = tan \(\theta\)

Squaring on both sides,

(\(\sqrt 2\) sin\(\theta\))² = (tan\(\theta\))²

or, 2 sin²\(\theta\) = tan²\(\theta\)

or, 2 sin²\(\theta\) = \(\frac {sin^2\theta}{cos^2\theta}\)

or, 2 sin²\(\theta\) cos²\(\theta\) = sin²\(\theta\)

or, 2 sin²\(\theta\) cos²\(\theta\) - sin²\(\theta\) = 0

or, sin²\(\theta\) (2 cos²\(\theta\) - 1) = 0

Either,

sin²\(\theta\) = 0

sin\(\theta\) = 0

sin\(\theta\) = sin 0°, sin 180°

Or,

2 cos²\(\theta\) - 1 = 0

2 cos²\(\theta\) = 1

cos²\(\theta\) = \(\frac 12\)

cos\(\theta\) =± \(\frac 1{\sqrt 2}\)

Taking Positive sign,

cos\(\theta\) = cos 45°, cos (360° - 45°)

Taking Negative sign,

cos\(\theta\) = cos (180° - 45°), cos (180° + 45°)

∴ \(\theta\) = 0°, 45°, 135°, 135°, 225°, 315° _Ans

Here,

\(\sqrt 3\) sin\(\theta\) - \(\sqrt 2\) = cos\(\theta\)

or, \(\sqrt 3\) sin\(\theta\) - cos\9\theta\) = \(\sqrt 2\)

or, \(\frac {\sqrt 3 sin\theta}{\sqrt {{\sqrt {(3)^2}} + (-1)^2}}\) -\(\frac {cos\theta}{\sqrt {{\sqrt {(3)^2}} + (-1)^2}}\) =\(\frac {\sqrt 2}{\sqrt {{\sqrt {(3)^2}} + (-1)^2}}\)

or, \(\frac {\sqrt 3 sin\theta}{\sqrt {3 + 1}}\) -\(\frac {cos\theta}{\sqrt {3 + 1}}\) =\(\frac {\sqrt 2}{\sqrt {3 + 1}}\)

or, \(\frac {\sqrt 3}{\sqrt 4}\) sin\(\theta\) - \(\frac {cos\theta}{\sqrt 4}\) = \(\frac {\sqrt 2}{\sqrt 4}\)

or, \(\frac {\sqrt 3}2\) sin\(\theta\) - \(\frac 12\) cos\(\theta\) = \(\frac {\sqrt 2}2\)

or, cos 30° sin\(\theta\) - sin30° cos\(\theta\) = \(\frac {\sqrt 2}{\sqrt 2 × \sqrt 2}\)

or, sin\(\theta\) cos 30° - cos\(\theta\) sin 30° = \(\frac 1{\sqrt 2}\)

or, sin (\(\theta\) - 30°) = sin 45°, sin 135°

Either,

\(\theta\) - 30° = 45°

\(\theta\) = 45° + 30°

\(\theta\) = 75°

OR,

\(\theta\) - 30° = 135°

\(\theta\) = 135° + 30°

\(\theta\) = 165°

∴ \(\theta\) = 75° , 165° _Ans

Here,

sin 3\(\theta\) + sin\(\theta\) = sin 2\(\theta\)

or, 2 sin\(\frac {3\theta + \theta}2\) cos\(\frac {3\theta - \theta}2\) - sin 2\(\theta\) = 0

or, 2 sin\(\frac {4\theta}2\) cos\(\frac {2\theta}2\) - sin 2\(\theta\) = 0

or, 2 sin 2\(\theta\) cos\(\theta\) - sin 2\(\theta\) = 0

or, sin 2\(\theta\) (2 cos\(\theta\) - 1) = 0

Either,

sin 2\(\theta\) = 0

2 sin\(\theta\) cos\(\theta\) = 0

or, sin\(\theta\) = 0

or, sin\(\theta\) = sin 0°, sin 180°

\(\theta\) = 0° , 180°

or, cos\(\theta\) = 0

or, cos\(\theta\) = cos 90°

\(\theta\) = 90°

or, 2 cos\(\theta\) - 1 = 0

or, 2 cos\(\theta\) = 1

or, cos\(\theta\) = \(\frac 12\)

or, cos\(\theta\) = cos 60°

\(\theta\) = 60°

∴ \(\theta\) = 0°, 60°, 90° and 180° _Ans

Here,

\(\sqrt 3\) sin\(\theta\) + cos\(\theta\) = 1

\(\sqrt {\sqrt {(3)^2} + (1)^2}\) = \(\sqrt {3 + 1}\) = \(\sqrt 4\) = 2

Dividing by 2 on both sides of the given equation,

\(\frac {\sqrt 3}2\) sin\(\theta\) + \(\frac 12\) cos\(\theta\) = \(\frac 12\)

or, sin\(\theta\) \(\frac {\sqrt 3}2\) + cos\(\theta\) \(\frac 12\) = \(\frac 12\)

or, sin\(\theta\) cos 30° + cos\(\theta\) sin 30° = \(\frac 12\)

or, sin (\(\theta\) + 30°) = sin 30°, sin (180° - 30°), sin (360° + 30°)

Either,

\(\theta\) + 30° = 30°

\(\theta\) = 30° - 30°

\(\theta\) = 0°

Or,

\(\theta\) + 30° = 150°

\(\theta\) = 150° - 30°

\(\theta\) = 120°

Or,

\(\theta\) + 30° = 390°

\(\theta\) = 390° - 30°

\(\theta\) = 360°

∴ \(\theta\) = 0°, 120°, 360° _Ans

Here,

\(\sqrt 3\) cos x + sin x

Dividing the equation by\(\sqrt {\sqrt {(3)^2} + (1)^2}\) = \(\sqrt {3 + 1}\) = \(\sqrt 4\) = 2

or, \(\frac {\sqrt 3}2\) cos x + \(\frac 12\) sin x = \(\frac {\sqrt 3}2\)

or, sin 60° cos x + cos 60° sin x = \(\frac {\sqrt 3}2\)

or, sin (60° + x) = sin 60°, sin (180° - 60°), sin (360° + 60°)

Either,

60° + x = 60°

x = 60° - 60°

x = 0°

Or,

60° + x = 120°

x = 120° - 60°

x = 60°

Or,

60° + x = 420°

x = 420° - 60°

x = 360°

∴ x = 0°, 60°, 360° _Ans

Here,

cos 3x + cos x = 2 cos x

or, 2 cos\(\frac {3x + x}2\) cos\(\frac {3x - x}2\) = 2 cos x

or, 2 cos 2x cos x = 2 cos x

or, 2 cos 2x cos x - 2 cos x = 0

or, 2 cos x (cos 2x - 1) = 0

or,cos x (cos 2x - 1) = 0

Either,

cos x = 0

cos x = cos 90°, cos 270°

Or,

cos 2x - 1 = 0

cos 2x = 1

cos 2x = cos 0°, cos 360°

2x = 0° , 360°

x = 0° , 180°

∴ x = 0°, 90°, 180°, 270° _Ans

Here,

\(\sqrt 3\) sin\(\alpha\) - cos\(\alpha\) = \(\sqrt 2\)

or, \(\frac {\sqrt 3 sin\alpha}{\sqrt {{\sqrt {(3)^2}} + (-1)^2}}\) -\(\frac {cos\alpha}{\sqrt {{\sqrt {(3)^2}} + (-1)^2}}\) =\(\frac {\sqrt 2}{\sqrt {{\sqrt {(3)^2}} + (-1)^2}}\)

or, \(\frac {\sqrt 3 sin\alpha}{\sqrt {3 + 1}}\) -\(\frac {cos\alpha}{\sqrt {3 + 1}}\) =\(\frac {\sqrt 2}{\sqrt {3 + 1}}\)

or, \(\frac {\sqrt 3}{\sqrt 4}\) sin\(\alpha\) - \(\frac {cos\alpha}{\sqrt 4}\) = \(\frac {\sqrt 2}{\sqrt 4}\)

or, \(\frac {\sqrt 3}2\) sin\(\alpha\) - \(\frac 12\) cos\(\alpha\) = \(\frac {\sqrt 2}2\)

or, cos 30° sin\(\alpha\) - sin30° cos\(\alpha\) = \(\frac {\sqrt 2}{\sqrt 2 × \sqrt 2}\)

or, sin\(\alpha\) cos 30° - cos\(\alpha\) sin 30° = \(\frac 1{\sqrt 2}\)

or, sin (\(\alpha\) - 30°) = sin 45°, sin 135°

Either,

\(\alpha\) - 30° = 45°

\(\alpha\) = 45° + 30°

\(\alpha\) = 75°

OR,

\(\alpha\) - 30° = 135°

\(\alpha\) = 135° + 30°

\(\alpha\) = 165°

∴ \(\alpha\) = 75° , 165° _Ans